scholarly journals Properties of the Reed-Muller spectrum of symmetric functions

2007 ◽  
Vol 20 (3) ◽  
pp. 281-294
Author(s):  
Claudio Moraga ◽  
Radomir Stankovic
Keyword(s):  
Boolean Functions ◽  
Symmetric Functions ◽  
Spectral Domain

Different forms of symmetry based on cofactors of Boolean functions are characterized in the Reed Muller spectral domain. Furthermore it is shown that if the arguments of the function are reordered, the permutation that is needed on the truth vector applies also on the spectrum of the function.

scholarly journals Designing Plateaued Boolean Functions in Spectral Domain and Their Classification

2019 ◽  
Vol 65 (9) ◽  
pp. 5865-5879 ◽  
Author(s):  
Samir Hodzic ◽  
Enes Pasalic ◽  
Yongzhuang Wei ◽  
Fengrong Zhang
Keyword(s):  
Boolean Functions ◽  
Spectral Domain

scholarly journals Logic Synthesis for a Regular Layout

VLSI Design ◽  
1999 ◽  
Vol 10 (1) ◽  
pp. 35-55 ◽  
Author(s):  
Malgorzata Chrzanowska-Jeske ◽  
Yang Xu ◽  
Marek Perkowski
Keyword(s):  
Boolean Functions ◽  
Symmetric Functions ◽  
Logic Synthesis ◽  
Regular Structure ◽  
Theoretical Background ◽  
Device Performance ◽  
Decision Diagrams ◽  
Binary Decision ◽  
Function Representation ◽  
New Algorithms

New algorithms for generating a regular two-dimensional layout representation for multi-output, incompletely specified Boolean functions, called, Pseudo-Symmetric Binary Decision Diagrams (PSBDDs), are presented. The regular structure of the function representation allows accurate prediction of post-layout areas and delays before the layout is physically generated. It simplifies power estimation on the gate level and allows for more accurate power optimization. The theoretical background of the new diagrams, which are based on ideas from contact networks, and the form of decision diagrams for symmetric functions is discussed. PSBDDs are especially well suited for deep sub-micron technologies where the delay of interconnections limits the device performance. Our experimental results are very good and show that symmetrization of reallife benchmark functions can be done efficiently.

scholarly journals The Weight and Nonlinearity of 2-rotation Symmetric Cubic Boolean Function

2015 ◽  
Vol 7 (2) ◽  
pp. 187 ◽  
Author(s):  
Hongli Liu
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Symmetric Functions ◽  
Recursive Formula ◽  
Symmetric Boolean Function ◽  
Symmetric Boolean Functions ◽  
Rotation Symmetric ◽  
Rotation Symmetric Boolean Function ◽  
Rotation Symmetric Boolean Functions

The conceptions of $\chi$-value and K-rotation symmetric Boolean functions are introduced by Cusick. K-rotation symmetric Boolean functions are a special rotation symmetric functions, which are invariant under the $k-th$ power of $\rho$.In this paper, we discuss cubic 2-value 2-rotation symmetric Boolean function with $2n$ variables, which denoted by $F^{2n}(x^{2n})$. We give the recursive formula of weight of $F^{2n}(x^{2n})$, and prove that the weight of $F^{2n}(x^{2n})$ is the same as its nonlinearity.

scholarly journals On the Sensitivity Complexity of k -Uniform Hypergraph Properties

2021 ◽  
Vol 13 (2) ◽  
pp. 1-13
Author(s):  
Qian Li ◽  
Xiaoming Sun
Keyword(s):  
Boolean Functions ◽  
Symmetric Functions ◽  
Uniform Hypergraph

In this article, we investigate the sensitivity complexity of hypergraph properties. We present a k -uniform hypergraph property with sensitivity complexity O ( n (⌈ k/3 ⌉) for any k ≥ 3 , where n is the number of vertices. Moreover, we can do better when k ≡ 1 (mod 3) by presenting a k -uniform hypergraph property with sensitivity O (n⌈ k/3 ⌉-1/2). This result disproves a conjecture of Babai, which conjectures that the sensitivity complexity of k -uniform hypergraph properties is at least Ω ( n k/2 ). We also investigate the sensitivity complexity of other symmetric functions and show that for many classes of transitive Boolean functions the minimum achievable sensitivity complexity can be O (N 1/3 ), where N is the number of variables.

A note on quantum algorithms and the minimal degree of epsilon-error polynomials for symmetric functions

2008 ◽  
Vol 8 (10) ◽  
pp. 943-950
Author(s):  
R. de Wolf
Keyword(s):  
Complexity Theory ◽  
Boolean Functions ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Quantum Algorithms ◽  
Minimal Degree ◽  
Close Connection ◽  
Worst Case

The degrees of polynomials representing or approximating Boolean functions are a prominent tool in various branches of complexity theory. Sherstov recently characterized the minimal degree $deg_{\eps}(f)$ among all polynomials (over $\mathbb{R}$) that approximate a symmetric function $f:\01^n\rightarrow\01$ up to worst-case error $\eps$: $ deg_{\eps}(f)=\widetilde{\Theta}\left(deg_{1/3}(f) + \sqrt{n\log(1/\eps)}\right).$ In this note we show how a tighter version (without the log-factors hidden in the $\widetilde{\Theta}$-notation), can be derived quite easily using the close connection between polynomials and quantum algorithms.

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